The answer is . Search google with "torsion coefficients" or "invariant factors" for finitely generated abelian groups. I'll give another example. For , you can decompose like,

.

Torsion coefficients (multiply numbers in each column) are 2, 12, 60 where 2|12 and 12|60.

Let , where . You can check that f is a surjective group homomorphism. Now the kernel of f is , the result follows from the first isomorphism theorem.2. Let F be a field.

Prove that

I realy need your guidance in these ones.

Thanks in advance